The Prime Minister questioned the corporation’s actions after an inquiry found the broadcaster covered up “deceitful behaviour” used by Bashir to secure his headline-making interview in 1995.
Speaking during a visit to Portsmouth on Friday, Mr Johnson warned steps needed to be taken to prevent something like it happening again.
He said: "I'm obviously concerned by the findings of Lord Dyson's report - I'm very grateful to him for what he has done.
“I can only imagine the feelings of the royal family and I hope very much that the BBC will be taking every possible step to make sure nothing like this ever happens again.”
The 127-page report found Bashir was in “serious breach” of the BBC’s producer guidelines when he faked bank statements and showed them to Diana’s brother Earl Spencer to gain access to the princess.
Senior BBC executives were also criticised in the report over a 1996 internal investigation which examined the mocked-up documents relating to the earl’s former employee.
It said: "I have concluded that, without justification, the BBC covered up in its press logs such facts as it had been able to establish about how Mr Bashir secured the interview; and failed to mention the issue at all on any news programme and thereby fell short of the high standards of integrity and transparency which are its hallmark."
Now Scotland Yard, which previously said it would not launch a criminal investigation into Bashir’s actions, has revealed it will “assess” the report to “to ensure there is no significant new evidence”.
William and Harry have also condemned the BBC for its treatment of Diana, claiming their mother’s Panorama interview fuelled her “fear, paranoia and isolation” and a wider “culture of exploitation and unethical practices ultimately took her life”.
In his rebuke of the BBC, the Duke of Cambridge said: “The interview was a major contribution to making my parents’ relationship worse and has since hurt countless others.
“It brings indescribable sadness to know that the BBC’s failures contributed significantly to her fear, paranoia and isolation that I remember from those final years with her.
“But what saddens me most, is that if the BBC had properly investigated the complaints and concerns first raised in 1995, my mother would have known that she had been deceived.”
Now William has demanded the documentary never be aired again.
The 38-year-old said: “In an era of fake news, public service broadcasting and a free press have never been more important.
“These failings, identified by investigative journalists, not only let my mother down, and my family down; they let the public down too.”
The UK Justice Secretary has said ministers will be looking into whether there were BBC governance issues outside of the remit of Lord Dyson’s reports that needed reviewing.
Robert Buckland said: “My colleague the Culture Secretary, Oliver Dowden, has rightly said that we should look at the governance structures of the BBC.
“We will take time to do that – the report that Lord Dyson issued yesterday is 127 pages long, so that needs to be looked at very carefully.
“And there may be issues that Lord Dyson wasn’t asked to cover that need to be looked at more widely, so it is a very serious moment for the BBC.
“They have apologised, which is appropriate, but clearly the wider issues of governance and the way things are run now need to be looked at.”
The scandal has also prompted a rebuke from media watchdog Ofcom, which warned further action may be necessary.
Dame Melanie Dawes, Ofcom’s chief executive, said: “Lord Dyson’s findings are clearly of great concern and raise important questions about the BBC’s transparency and accountability.
"As the BBC’s independent regulator Ofcom is considering the report, and will be discussing with the BBC what further actions may be needed to ensure that this situation can never be repeated.”
Dame Melanie also revealed former BBC executive Tim Suter, who was part of the 1996 internal investigation into Martin Bashir’s interview with Diana, Princess of Wales, has stepped down from his current board role with Ofcom.
She added: “By mutual agreement, Tim Suter, Ofcom board member and chair of Ofcom’s content board, is stepping down with immediate effect.
"We would like to thank Tim for his contribution to Ofcom.”
Former director of BBC News James Harding said he took responsibility for the rehiring of Martin Bashir by the corporation in 2016.
Mr Bashir left the BBC in 1999, before returning in 2016 and becoming religion editor.
Mr Harding told BBC News: “It’s distressing for everyone, and it’s depressing for anyone who cares about journalism and the BBC.
“I wanted to say so much of what is known now was not known then, and certainly not by me.”
When asked if he knew that Mr Bashir had forged bank statements, he replied: “I didn’t know and in fact if I had he wouldn’t have got the job.”
Mr Harding refused to comment on whether he had discussed rehiring Mr Bashir with then director-general Tony Hall, saying instead: “The way I think about it is that I was running BBC when Martin Bashir was hired back into BBC News, and so the responsibility for that sits with me.”
Mr Bashir has now apologised again for his actions.
He said: "This is the second time that I have willingly fully co-operated with an investigation into events more than 25 years ago.
"I apologised then, and I do so again now, over the fact that I asked for bank statements to be mocked up.
"It was a stupid thing to do and was an action I deeply regret."